On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres. III (Q1058152)

This is a continuation of two previous papers [see ibid. 18, 115--145 (1981; Zbl 0482.53046) and the preceding review Zbl 0564.53022]. In this paper we study the spectrum of the Jacobi differential operator \(\tilde S\) for minimally immersed spheres into spheres. Computing the matrix expressions of the linear mapping \(S_{\sigma}\), defined in subsection 5.2 of our first paper, we show that every eigenvalue of the Jacobi differential operator \(\tilde S\) is an algebraic number, however not a rational number in general. This suggests that \(\tilde S\) will not be described only by Casimir operators. We give a lower bound for the nullity of \(\tilde S\). In particular, for the minimally immersed 2-dimensional sphere \(S^ 2\), the nullity is explicitly computed and we show that the nullity is equal to twice the Killing nullity.